Explanation of number base conversion

Normally we express numbers in base ten, however it is possible to use any positive integer greater than one as the base for numbers. Other systems that are popular are the binary, octal, and hexadecimal systems.

When we write a number in base ten, for example 376; the number is expressed in terms of powers of ten like this 376 = 3*ten^2 + 7*ten + 6. The positions of the digits tells the power of ten multiplied by that digit. In other systems the base is no longer ten, but the idea is the same.

In base five, for example, we have only the digits 0, 1, 2, 3, and 4. Six has to be expressed as 10, seven as 11, and eight as 12; since six = 1*five + 1, 7 = 1*five + 2, and eight = 1*five + 3.

Still in base five we have the number 1433. What is this number in base ten? Well, 1433(base five) = 1*five^3 + 4*five^2 + 3*five + 3. You can work this out as 125 + 4*25 + 3*5 + 3 = 243.

Octal numbers (base eight) are written using eight distinct symbols: 0, 1, 2, 3, 4, 5, 6, and 7. Once we have counted up to 7 we run out of digits, so we must place a 1 in the "eights' column." Counting from one to twenty in base eight goes like this: 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 22, 23, 24. Find 71(base ten) expressed in base eight. The answer is 107, since 71 = 1*eight^2 + 0*eight + 7.

Binary numbers (base two) use only 0 and 1. Counting from 1 to 10 in binary we have 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010. Here we run out of digits very soon -- after counting to 1 infact. Notice that the powers of two -- 2, 4, 8, 16, etc. -- are represented in base two by 10, 100, 1000, 10000, and so on -- just as the powers of ten are in base ten notation.